Do I need a building permit to add a side door and back deck to my home in a single-residential neighborhood? If so, where would I get the permit?

Most likely YES... call your town or County Zoning office.

Typically you don't just "get" a permit. You file an application for one, add details as to what you are planning to do. For example, they will need to know if the door will be going in a bearing or non bearing wall, how high the deck will be, what size, etc. Decks are built to specific span tables, size, how you attach to house etc. After they agree to the details, you will pay for the permit and they will inspect.